where \(q,r\)  is a complex-valued function of \(x\)  and \(t\) ,
has been studied extensively in
relation with various aspects such as Lax integrability, bi-Hamiltonian
structure, various kinds of exact solutions, etc.[1]}, [2]}, [3]}, [4]}, [5]}, [6]}.
In ref. [6]}, we have constructed the FL hierarchy using polynomial recursion formalism,
then derived the algebro-geometric solutions of the whole hierarchy
and degenerated them into
\(n\) -dark solitons.
To find solutions we go through an intermediate step,
a complexified version of FL hierarchy.
However,
in physical applications,
two natural reductions of (REF ),
the defocusing case, with \(r=-\overline{q},\)
\(\textrm {FL}_-:\quad q_{xt}-q_{xx}-i|q|^2q_x-2i q_x+q=0,\) 
